Representation of a target Hamiltonian (left) and its associated gadget Hamiltonian, as defined in Eq. (3), for the case of n=8, k=4, and r=3. Qubits that interact according to the terms of the Hamiltonian are placed in the same colored region. Shown in the central and right-hand panels are the contributions to the gadget Hamiltonian from two of the three four-body terms, with the auxiliary qubits in gray and the target qubits in black.